First of all there is a crucial point that we have to clarify. In many case also econometrics books are ambiguous about that and probably you refers on. The main assumption requested is like $E[\epsilon|X] = 0$ named exogenous condition. The concept of exogeneity is related to causal inference, large part of econometrics problems are referred on it. Unfortunately causal inference in econometrics are badly treated, as pointed out in: Chen and Pearl (2013). Many relevant problems are revealed by this article but in my opinion some others are not adequately addressed. Those are mostly related to the concept of true model. This concept is very used in econometrics literature but, almost never enough words are spent about it in the books. Often almost nothing is say about it.
In particular the crucial point is: the true model is write down like a regression, in most case linear, but it is not a regression (linear or not). True model is something else. In my opinion the best way to think about it is as a structural causal model, in most case linear structural causal model.
These discussions are related
Regression and causality in econometrics
Difference Between Simultaneous Equation Model and Structural Equation Model
What is a 'true' model?
Now, you unambiguously talk about OLS regression equation like:
$Y = \alpha + X’ \beta + \epsilon$
Note that the term $\epsilon$ is a residual.
Then in this situation your assumption 2 are useless because the uncorrelatedness requested holds, in any case, by construction not by assumption. Often this uncorrelatedness is erroneously conflated with a sort of weak form of exogeneity.
Statistically speaking, mean independence, also holds by construction if the exact conditional expectation function is linear. If all the variables involved (dependent and independent) have jointly Normal distribution, also stochastic independence holds by construction. However, disentangled from improper conflations, nothing of the above say something about exogeneity in its proper meaning.
Therefore, in any case, even if homoscedasticity can be an assumption yet, your question is senseless.
Now, if we consider the same equation as before like a true model we have to note that the term $\epsilon$ is no more a residual but a structural error term; completely another thing. Here your question become well posed. So, from homoscedasticity we have that:
$V[Y|X]= V [ \alpha + X’ \beta + \epsilon |X] = V [ \epsilon |X] = \sigma^2$ and
$V [ \epsilon |X] = E[\epsilon^2|X] – (E[\epsilon|X])^2 = \sigma^2 $
Actually this imply that $E[\epsilon|X]=c$, however we cannot sure that this constant term is $c=0$. Infact we have to note that, different than in regression, nothing is true by construction about $\epsilon$.
Finally if we assume homoscedasticity and $E[\epsilon]=0$ also, after all a natural features for any kind of error term, the reply to your question is yes. Uncorrelatedness between dependent variables and error term is implied; mean independence also. In other words, exogeneity is implied.
Actually this seems me an interesting result and I don't know why no author underscored it. However seem me that authors have greater problems to solve.
Moreover can be useful to say that in regression analysis heteroscedasticity can reveal misspecification problems. Therefore homoscedasticity is a good thing.
